628 TRANSACTIONS OF SECTION A. 



A function V = F(a\ y, z, w) when expressed a8 a homogeneous function of 

 degree zero in (I, m, n. X, fi, v) will be a solution of 



dl d\ dm dfi dn dv 



when expressed as a homogeneous function of degree — 1 in (l, m, n, X, fi,, r) it will 

 be a solution of 



a-^F a^F d''F d-V 



d3:~ dy'^ dz^ die- 



9^9x 9?w3fi dndv 



A particular solution of the last equation, which is a homogeneous function of 

 degree — 1 , is given by 



(a h c \ 

 a'/3'y' ; 

 where P is Riemann's general hypergeometric function and 



Ik^{b-c){6~a), mn = (c-a){6-b), nv={a-h){6-c) 

 o + a' + )3 + /3'+y + 'y' = ]. 



The group of conformal transformations is derived from the group of auto- 

 morphic linear transformations of the coordinates (/, m, n, X, fi, v) in which the 

 expression 



l\ + mp. + nv 



is left unaltered in form. 



By putting w = ivt, where v is the velocity of light, we may obtain a number of 

 transformations which can be applied to optical problems. Of these we shall 

 mention two : — 



and 



v — V — y 7 _ ^ 'p — 



r' — v^t' r — v't' v — vt- i- — v'v 



vt' z-vV " 2(= - vt) ' ~ 2(s - It) 



In either case if F(X, Y, Z, T) is a solution of 



8^F 8=F^ a^F _ 1 ^ 



ax'^ + ay- "^ az-' " v' ax^ 



so that when F = is solved for T, the function T is the characteristic function of 

 a series of parallel wave surface."^, the transformation gives us an expression for F 

 in terms of (r, y, s, t), which is also such that if the equation F = be solved 

 for t, then t is the characteristic function for a second system of parallel surfaces. 

 In the case of the first transformation the surface ^ = is the inverse of the 

 correfponding surface T = 0. This transformation may be studied further by 

 supposing a point (r, y, z) to start at a point (|, 17, () on a surface /, and to move 

 with uniform velocity v along a straight line whose direction cosines are (/, m, n), 

 so that its coordinates at time t are given by 



x=-^+ h't, y = l + m vt, = = C + ''"■^- 



